Maryanthe Malliaris and Saharon Shelah. Cofinality spectrum problems in model theory, set theory and general topology. J. Amer. Math. Soc., vol. 29 (2016), pp. 237-297. Maryanthe Malliaris and Saharon Shelah. Existence of optimal ultrafilters and the fundamental complexity of simple theories. To appear. Advances in Mathe- matics, arXiv:1404:2919[math.LO]. Maryanthe Malliaris and Saharon Shelah. Keisler’s order has infinitely many classes. arXiv:1503.08341[math.LO].
In this ground-breaking series of papers, longstanding open problems in both model theory and set theory are solved. The authors indicate that these papers are part of an ongoing program. In this review we give a snapshot of the current state of the program. For the past fifty years, one of the dominant themes in model theory has been Shelah’s classification program, which seeks to classify first order theories according to how ‘tame’ the models of the theory are. The set of stable theories has been classified in a coherent way using the Morley rank and other concepts, and is well understood. A major goal of current research in model theory is to achieve a similar understanding of the unstable theories. In the paper [H. Jerome Keisler, Ultraproducts which are not saturated, Journal of Symbolic Logic vol. 32 (1967), 23–46] I introduced a pre-ordering E on all complete first order theories, that measures the degree of reluctance of ultrapowers of models of a theory to be saturated. In that paper I posed the problem of determining the structure of this ordering, and asked whether it would give a fruitful classification of first order theories. This problem has turned out to be extraordinarily difficult, and little progress had been made until the recent work of Malliaris and Shelah. The papers under review show that the ordering E gives a new and unexpected classification of the simple theories. They also solve a major problem in set theory that had been open since the 1940’s by showing that p = t. It is quite surprising that this is actually a consequence of ZFC rather than an independence result, and that it comes out of a model-theoretic study of the E ordering. Here is a brief review of the situation before about 2010. We use T, U, . . . for complete first order theories with countable signatures, D for an ultrafilter over an infinite set I, and λ for the cardinality of I. By the fundamental theorem ofLo´s,every ultrapower of a model of T is a model of T . We say that D saturates T (or is good for T ) if the ultrapower M I /D is λ+-saturated for every model M of T . Two important properties of filters are regularity and goodness. A filter F over I is regular if there is a set E ⊆ F of cardinality λ such that each i ∈ I belongs to only finitely many e ∈ E. F is good if every monotonic function f :[λ]<ℵ0 → F has a multiplicative refinement g :[λ]<ℵ0 → F (that is, g(u) ⊆ f(u) and g(u ∩ v) = g(u) ∩ g(v)). In the early 1960’s I introduced good filters, and showed that a regular ultrafilter D is good if and only if D saturates every T , and that the GCH implies that regular good ultrafilters exist. In 1972, Kunen proved the existence of regular good ultrafilters in ZFC. In the 1967 paper referenced above, T E U is defined to mean that for every infinite cardinal λ, every regular ultrafilter over a set of power λ that saturates U also saturates T . T/U means T E U and not U E T . The relation E is a pre-ordering of the class of all complete theories, and induces a partial ordering on the set of E-equivalence classes of theories. The following results are from that paper. If D is regular and M I /D is λ+-saturated for some model of T , then D saturates T . There is a theory T that is E-minimal in the sense that T E U for all U, and a theory U that is E-maximal in the sense that T E U for all T . If T is E-minimal, then T does not have the finite cover property (fcp) and is not E-maximal.
1 2
Shelah proved the following results in the 1970’s; details can be found in Chapter VI of the book [Saharon Shelah, Classification Theory, North-Holland 1990]. The minimal E-equivalence class is the set all theories that do not have the fcp, and every such theory is stable. The next lowest E-equivalence class is the set of stable theories with the fcp. If T is stable and U is unstable then T/U. Every theory with the strict order property is E-maximal. There was little further progress for the next several decades. Because of these early results, people suspected that the ordering E would be coarse, and that there may only be a small finite number of E-equivalence classes. The recent papers by Malliaris and Shelah contain the following results about E, which completely change the picture. (1) There is a minimum E-equivalence class of unstable theories. It contains the theory Trg of the random graph. (2) If T is not low or not simple, then Trg /T . Thus every theory that is E-equivalent to Trg is low, simple, and unstable. (3) There is an infinite sequence of theories (U0,U1,...,Un,... ) such that
Trg /.../Un /.../U1 /U0,
and for each n, every theory that is E-equivalent to Un is low and simple. (4) Suppose there is a supercompact cardinal. If T is simple and U E T , then U is simple. (5) There is a minimum E-equivalence class of non-simple theories. It contains the ∗ theory Tfeq, which is the model completion of the theory of an infinite family of independent parametrized equivalence relations. (6) Every SOP2 theory is E-maximal. It follows from (1)–(3) that the set of E-equivalence classes of simple unstable theories has a E-minimum element, and is infinite and not well-ordered by E. Result (1) is from the paper [Malliaris, Hypergraph sequences as a tool for saturation of ultrapowers, Journal of Symbolic Logic vol. 77 (2012), pp. 195-223]. Result (2) is from the paper [Malliaris and Shelah, A dividing line within simple unstable theories, Advances in Mathematics vol. 249 (2013), pp. 250-288]. Result (3) is partly in the second and partly in the third paper under review. (4) is in the second paper under review. (5) and (6) are in the first paper under review. In order to obtain the results (1)–(6), Malliaris and Shelah introduced new model- theoretic notions involving the realization and omitting of types, and new set-theoretic properties of ultrafilters. The latter are often “refinement properties”—weakenings of goodness that require only that certain monotonic functions have multiplicative refinements. Powerful methods were developed to construct ultrafilters, and to relate properties of ultrafilters to properties of ultrapowers. The following paragraphs will describe some of these methods in broad terms, skipping many details. From now on, D will always denote a regular ultrafilter. A basic method in the proofs of results (2)–(4) is the construction of a regular ultrafilter over a set I from a (not necessarily regular) ultrafilter on a complete Boolean algebra B. Let D∗ be an ultrafilter on B. Say that D is built from D∗ if there is −1 a surjective homomorphism j : P(I) → B such that j ({1B}) is a good regular filter −1 over I, and D = j (D∗). A general result, called “separation of variables”, shows that an ultrafilter D∗ on B has a refinement property called moral for T if and only if every D built from D∗ saturates T . This is useful because one often has more freedom in constructing ultrafilters on B than over I.
Given cardinal numbers θ ≤ µ < λ, B2λ,µ,θ is the completion of the Boolean algebra λ freely generated by elements xf where f is a function whose domain is a subset of 2 of size < θ, subject to the conditions that xf ≤ xg when f ⊆ g, and xf ∩ xg = 0 when 3 f, g are incompatible. It is shown that for every ultrafilter D∗ on B2λ,µ,θ, there is a −1 surjective j such that j ({1B}) is a good regular filter over I, and hence there exists a regular D that is built from D∗ (2) is related to a refinement property that has been around since the 1960’s: D is OK if every monotone function f :[λ]<ℵ0 → D, such that f(u) = f(v) whenever |u| = |v|, has a multiplicative refinement g :[λ]<ℵ0 → D. In the above (2012) paper, Malliaris introduced a refinement property called flexibility, which was later found to be equivalent to being OK. She showed that if D saturates a theory that is not low or µ not simple, then D is flexible. Let ω ≤ µ < λ ≤ 2 and let B = B2λ,µ,ω. In the above (2013) paper, it is shown that no ultrafilter D that is built from an ultrafilter on B is flexible, and that there is an ultrafilter D∗ in B that is moral for Trg. Hence any D that is built from D∗ saturates Trg, and any theory that is saturated by D is low and simple. Then for any theory U that is not low or not simple, not Trg E U. We also have Trg E U by (1), so Trg /U. (3) Let Tm,k be the model completion of the theory with one symmetric irreflexive (k + 1)-ary relation with no complete subgraphs on m + 1 vertices. Hrushovski showed that Tm,k is a low simple theory when m > k ≥ 2. For each n, let Un be the disjoint union of the theories Tk+1,k for k ≥ 2n+2. It is shown that (3) holds for these particular theories Un. To do this, the notion of a perfect ultrafilter is introduced. Being (λ, µ)-perfect is a refinement property of an ultrafilter D∗ on the complete Boolean algebra B = B2λ,µ,ω. Suppose α is an ordinal, 2 ≤ k < `, µ = ℵα, and λ = ℵα+`. The following results are proved. There exists a (λ, µ)-perfect ultrafilter D∗ on B. For every (λ, µ)-perfect ultrafilter D∗ on B and every m > `, D∗ is moral for Tm+1,m. But for every ultrafilter D∗ on B, D∗ is not moral for Tk+1,k. So taking ` = k + 1, we get D∗ that is moral for Tm+1,m for all m > k + 1, but not for Tk+1,k. Then any D that is built from D∗ will saturate Tm+1,m for all m > k + 1, but will not saturate Tk+1,k. From the proof of (2) above, every theory that D saturates will be low and simple. It is easily seen that a regular ultrafilter will saturate Un if and only if it saturates Tk+1,k for every k ≥ 2n + 2. The result (3) follows. (4) Assume that σ is an uncountable supercompact cardinal. It is shown that there is a regular ultrafilter D that saturates the simple theories and only the simple theories.
This easily implies (4). D will be built from a σ-complete ultrafilter D∗ on B2λ,µ,σ. This is rather unexpected because a σ-complete ultrafilter can never be regular. We saw from the proof of (2) that an ultrafilter that is built from an ultrafilter on B2λ,µ,ω cannot be flexible, and hence cannot saturate a simple theory that is not low. This obstacle is avoided here by assuming that there is a supercompact cardinal σ and working with B2λ,µ,σ. Now assume that µ, λ are finite successor cardinals of σ with µ < λ. (The papers under review used a more general setting, carrying along a quadruple λ ≥ µ ≥ θ ≥ σ of cardinals satisfying a condition called suitability). Let B = B2λ,µ,σ. The model- theoretic property of a theory being (λ, µ, σ)-explicitly simple is introduced. It gets weaker as µ increases. Intuitively, µ is a bound on the complexity of amalgamation. A key result is that when λ = µ+,(λ, µ, σ)-explicit simplicity is equivalent to simplicity. (λ, µ, σ)-optimality is a refinement property for σ-complete ultrafilters on B that is a large-cardinal analogue of being (λ, µ)-perfect. The following results are proved. Every (λ, µ, σ)-optimal ultrafilter is moral for every (λ, µ, σ)-explicitly simple theory. 0 + There exists a σ-complete filter D∗ on B generated by µ sets such that no σ-complete 0 0 ultrafilter D∗ ⊇ D∗ is moral for a non-simple theory. Every such D∗ can be extended 0 to a (λ, µ, σ)-optimal ultrafilter D∗ ⊇ D∗. Hence for any theory T , D∗ is moral for T if and only if T is simple. So any D that is built from D∗ will be as required. 4
(5) is related to another refinement property of ultrafilters, called goodness for equal- ity. D is said to be good for equality if for every set A and every set X ⊆ AI of power |X| ≤ λ, there exists a mapping d : X → D such that for all x, y ∈ X, if the set {i ∈ d(x) ∩ d(y): x(i) = y(i)} is non-empty then it belongs to D. In the (2012) paper mentioned above, Malliaris showed that if D saturates a theory that is not simple and is not SOP2, then D is good for equality, and that if D is good for equality then D ∗ saturates Tfeq. The result (6) implies that if D saturates an SOP2 theory then D is ∗ good and hence also saturates Tfeq. Thus (6) implies (5). (6) In the proof of (6), ultrapowers of the model hN,
+ and (M1 , ∆) has enough set theory for trees in a natural sense. An example is the ultrapower CSP, in which I + + + I M = hN,
Therefore in V [G], C(s, t) = ∅. It is shown that in V [G], p < t implies that ps ≤ p, and hence ps ≤ t. This contradicts C(s, t) = ∅, so p = t holds in V [G]. Finally, this implies that p = t holds in V . The methods developed in these papers are likely to stimulate more research in model theory and set theory. An enticing possibility is that the general results on cofinality spectrum problems will have broader applications. Some definitions in the papers are quite complicated, and there will be a search for simpler alternatives. Many questions about the E-ordering remain open. The results on the E-ordering and on explicit simplicity will re-open work on the classification of simple theories. Keisler, H. Jerome Department of Mathematics,University of Wisconsin, Madison, WI, USA. [email protected].
Entry for the Table of Contents: Three papers by Malliaris and Shelah. Reviewed by Keisler, H. Jerome ...... xxx